Posted
by
Zonkon Friday December 21, 2007 @01:16PM
from the next-they'll-use-geometry-on-the-mystery-of-the-haunted-amusement-park dept.

mlimber writes "Do you ever find yourself in a traffic jam, thinking, 'Man, there must be a bad accident up ahead,' but as you plod along you see no evidence of any crash? Some mathematicians have solved the mystery by developing a mathematical model that shows how one driver hitting the brakes a little too hard can cascade into a backup miles behind. The mathematicians' future research will investigate how automatic braking systems may alleviate the problem."

I've often seen this. People slow down too much for no reason, especally near ramps. I've actually gotten pretty good and figuring which jams are accident related and those that are just people being retarded.

It doesn't help that speed limits on interstates get lowered as you approach larger cities. This is a good reason to remove enforced upper limits on these roads completely. Much of the braking is due to the few goody-goodies cramping the whole flow.

One of the most irritating driving habits I can think of are people who obsessively cover the brake every 20-30 seconds or so. Usually soccer moms in Suburbans or elderly in the largest Lincoln they could find. There's nothing ahead of them, no reason really to tap the peddle, but they do it anyway out of habit.

If an automatic braking system can solve this problem, I'm in for my tax dollars.

The difference is the overall level of asshole-ness of the drivers. Much of the braking in dense traffic is caused by desperate maneuvers. In New York, the maneuvers are caused by the fact that I'd say less than 10% of the drivers will allow anyone to merge ahead of them for ANY reason. In dense traffic moving at 50mph, it's not uncommon to be within 3-4 meters of the car in front of you, or even less. And since nobody will allow you to merge, you're forced to perform pretty daring high-acceleration maneuvers to force yourself into the target lane... which will cause that lane to rapidly decelerate, clearly creating the traffic wave.

All it would take to stop this from happening, is for people to stop being assholes, and to let you through, when you're trying to get into an exit, 1/4 mile away.

Not necessarily stupidity. if I can see past the driver in front of me, I can make a better decision. I can see that he is simply adjusting speed to allow for a more reasonable space between him and the car in front. If I am stuck behind an SUV in my car, then I am not sure if his tap on the brake is about to turn into a full fledged stomp of the brakes, and I have to adjust, and possible harder that I need to. This becomes a cascading event.

I'm with the rest of you. This is hardly a revelation. At Texas Motor Speedway, during a NASCAR weekend with 225,000 fans trying to leave, a person stopping for three seconds causes a 20-minute delay to the last car in line. Until they fixed the number of exits flowing back out to I-35, it usually took 3-4 hours to get out of the parking lot.

Another cause for bad traffic is the ridiculously easy driving test we have in the States. Couple that with law-enforcement only ticketing speeders instead of bad drivers in general, and you get the traffic we have in most of our cities. I also hate how all accidents are chalked up to "failure to control speed", which makes it sound as if speeding were the main cause of all accidents. In reality, failure-to-yield is overwhelmingly the #1 cause of collision accidents, not speed. But the revenue hungry cops would rather sit on their motorcycles with radar guns than actually pull people over for changing 5 lanes at once, or cutting off other drivers by pulling out in front of them and then NOT accelating.

Not to mention, hell will freeze over before they ever ticket a slow driver in the left lane.

The effect would be mitigated if everyone would refrain from following other cars so closely.

Yeah.... I tried this for several weeks. Except everyone took the opportunity to sneak in front of me so they could play a game of changing lanes repeatedly to snake their way through traffic faster.

That is the problem. You can do what is best for the group, but then selfish individuals abuse that for their own gain which hurts the group more. I can't wait until we have self driving cars... I could easily foresee traffic signals going away, much more efficient cars and no more worrying about getting to old to be safe on the roads. Add to this a dropped death rate, and this breakthrough would easily be the greatest advancement of the 21st century.

If I give 2 seconds to the car ahead, it is likely that two drivers and maybe a third idiot will wedge into that gap. Now I have to slow down to achieve the new two second gap, which will cause everyone behind me to react with breaking and more slowdowns. Eventually there will be a wave of breaking that causes a huge delay in traffic with no apparent cause. I may even be lucky enough to be run into from behind, and then 2 seconds at zero mph would be the exact lack of distance between our now entangled bumpers.

Now if we could actually give space to everyone and not have the self-righteous take advantage of these gaps as there way to shave 8 seconds off of their commutes you may have a point.

Sorry to be cynical to your point, but I live in Atlanta, and people here suck.

Yeah.... I tried this for several weeks. Except everyone took the opportunity to sneak in front of me so they could play a game of changing lanes repeatedly to snake their way through traffic faster.

It's funny how often I hear this. I try to hold back a safe distance all the time, and sure, a handful of morons weave through the gaps and I have to drop back a little more. But I never see this horrendous influx of morons people keep telling me about. I manage to maintain a much better distance (and a much smoother drive in terms of both vehicle speed and mental stress) than most people other than for a few moments if someone cuts in, and since those people usually cut out again almost as quickly, I doubt it even slows me down noticeably.

FWIW, this is the UK, and the comment above apply to both high-speed motorway driving and crowded conditions around the city. I've never driven in the US, but I do hear the same sorts of complaints the parent post was making all the time.

Very few of our motorways are actually equipped with this system, as far as I know it's only the middle of the M42 near Birminham and the Western 1/3 of the M25 London ringroad. As I have to drive on both of these occasionally, I am all too aware that while the idea might be sound, the implementation is hopeless. The M25 system is very basic, and unless you go there in the dead of night, you'll have to fight through exactly the sort of standing wave that this system is supposed to get rid of, caused by cars braking from the 70mph permitted outside the system to the 60 that the controlled section seems stuck at most of the time.

The M42 system is more complex, with limits enforced by hundreds of spy cameras, with the ability to allow cars to use the hard shoulder at busy times. This seems more of a revenue generating exercise than a congestion removal system, as the limit is nearly always 50mph even if there is barely another car in sight. Drivers have responded by speeding up between the camera traps and braking just before them, which is dangerous, causes exactly the standing wave effect it's designed to avoid, and wasted petrol increasing CO2 emissions... well done UK government!

This result is not only well known, it is classical. I'm sure there must
be some new twist to the analysis to make it publishable, but I can't find it.
The article made me search my bookshelves for an old textbook [amazon.com] I used for teaching a class in
Applied Mathematics back in the 80's (in my starving professor phase of life):

It has a very nice development of the math behind traffic flow, which also turn
out to be the 2D equations for a compressible fluid. And in a compressible fluid
the speed of flow decreases with density, causing the characteristic lines
(the direction of wave propagation) to cross, causing a classic shock wave to
form. The shock wave in traffic flow is the traffic jam. The section on Traffic
Flow is about 140 pages of the book. Select chapters names include:

Flow Equals Density Times Velocity

Conservation of the Number of Cars

Experimental Observations

Traffic Density Waves

After a Traffic Light Turns Green

Wave Propagation of Auto Brake Lights

Stationary Shock Waves

Effect of Red LIght or an Accident

and so on. It made for an interesting class (and the only math class where
I ever assigned a paper!). I recommend the book.
If only we could train people to behave in traffic as if they were an
incompressible fluid, we would never have traffic jams; but to do
that, you'd need to be trained to speed up when traffic got heavy.

Yeah.... I tried this for several weeks. Except everyone took the opportunity to sneak in front of me so they could play a game of changing lanes repeatedly to snake their way through traffic faster.

It's funny how often I hear this. I try to hold back a safe distance all the time, and sure, a handful of morons weave through the gaps and I have to drop back a little more. But I never see this horrendous influx of morons people keep telling me about. I manage to maintain a much better distance (and a much smoother drive in terms of both vehicle speed and mental stress) than most people other than for a few moments if someone cuts in, and since those people usually cut out again almost as quickly, I doubt it even slows me down noticeably.

FWIW, this is the UK, and the comment above apply to both high-speed motorway driving and crowded conditions around the city. I've never driven in the US, but I do hear the same sorts of complaints the parent post was making all the time.

It's not just the UK, I've noticed the same sort of thing in the US. This style of driving is also a lot more conducive to not getting bent out of shape.

It helps to realize that some people are assholes, and a lot of those assholes are not just bad drivers, they're dangerous to themselves and others. Keeping a safe distance instead of trying to block people probably isn't slowing you down and is definitely increasing your safety margins.

Don't forget drivers who do not use their turn signals. In some cases this is bad for traffic. There are many spots along my commute where I will be stopped at a Stop sign waiting for a car coming from the left to pass so I can keep going only for them to turn right. If they had used their signal, I could have proceeded. Instead traffic gets backed up at the Stop sign. This is true of a couple of rotaries I know as well.

Another problem I see every day is that of drivers who block intersections at a Red Light. I was pleased to finally see a cop ticketing for this recently.

I agree. People should be ticketed for these things. It would help traffic over-all.

So here is the problem, there are two classes of people on just about any road, anyplace:

Drivers

Motor Vehicle Operators

Drivers. These are the people you see driving, not overly fast, but driving with intent. They pay attention, they are generally never talking on a cell phone, their eyes are always scanning the road ahead, their mirrors and their instruments. They use blinkers AND turn them off, they can be pretty much any age and any gender. You will notice that they drive consciously.

Motor Vehicle Operators - These are the people you see driving a car that scare the crap out of you. They are NOT paying attention, they are shaving, eating, reading the paper, putting on makeup, doing their hair. Their cell is glued to their ear, are fiddling with the radio very three seconds. Their left turn blinker is invariably on.

Some things I would like to see tickets given for:

Merging onto the freeway at less then the speed of traffic

Changing lanes on the freeway into to an impossibly tight spot and then nailing the breaks. I don't mind if you are a driver and you do this, because if you are a driver, you do the maneuver with grace, authority, you have made sure the person you are going to be in front of knows your intent and keep up the pace.

Driving while eating

Driving with any sort of animal ( human or otherwise ) on your lap

Trying to light any sort of smoking material with a Bic lighter

Putting on makeup. Sorry girls, but if you shove the fucking mascara wand into your eye you will crash and probably kill yourself, but worse then that, kill someone else.

I think that should prime the pump, as I am sure my fellow/.'rs will add many many more.

I live in Los Angeles and commute 30 miles to work one way. It takes me an hour or more on most days, less at very early hours (pre 5AM). I've made a little mental hobby of traffic pattern prediction and on my route I can accurately predict where the slowdowns will be, and know which lanes to be in on the freeway in order to keep moving. I also know that the pattern changes from time to time and for various reasons and account for that as well.

By far the biggest problem with traffic here, other than the staggering number of people on the roads, is a false sense of entitlement and/or lack of courtesy for other drivers. I start my drive from a decent neighborhood and go through a pretty big slice of the city hitting East LA, Korea Town, West LA and downtown (including skid row and not in that order). It's not just soccer moms, it's not just the elderly, it's not just the Asians or the Latinos or the Blacks or the Whites or anything. It's ALL of them. For every decent driver out there, there's literally a thousand or more assholes. I moved here from Boston 10 years ago and I remember thinking "what's all this road rage shit I hear about?". How could you possibly get so worked up in your car that you'd want to KILL other drivers. Well I've seen it myself first hand out here.

About a month or so ago a Mom was killed and possibly her 2 kids as well (not sure) because 2 guys were fighting with each other on a busy surface street. One would hit the gas and then the brakes trying to get the other to rear end him or cutting the other off from getting in a lane or passing. Oh and by the way, yes one guy was about 19... but the other guy? He was in his 40s. You'd think after a certain *I'm invincible* phase people would grow up and mellow out. Most do, but some don't and some just want to go about their business, but when they're pushed, they push back. This is where I fit in. I mind my business and I try to drive quickly and efficiently without being too much of a jerk about cutting people off and I try to let people in when they need to. In other words, I *try* to be a courteous driver. If I'm in the fast lane out here with no one ahead of me, I'll be doing 90 easily, but if someone comes up behind me in a faster car, or just generally wants to drive faster than me, I'll move the fuck out of the way. I pay attention to my surroundings and I realize I'm not driving the fastest car on the road. Same applies no matter what lane I'm in on the freeway. I get the fuck out of the way, safely, efficiently and without waiting an hour. So few people do that here it's sad.

You say "drive the speed limit" its the law, it's there for a reason. I say, fuck off, I'll drive as fast as I think I can go safely. If I feel safe at 90, then I'm going to go 90. If I think it's safe at 40, I'm going to go 40. But I'm damned sure not going to BLOCK traffic or try to be the amateur police force by sitting in a lane, driving much slower than necessary and making it hard or next to impossible for anyone to get around me. I'm simply going to move OUT OF THE WAY. As for distances between cars, I try to leave plenty of room to stop, meaning at least 2 or 3 car lengths depending... BUT, here's the thing out here. You just CANNOT leave the 3 seconds or more of room that you'd like and still get anywhere. We're all not on a plane. We all don't *get there* at the same time. And yes, I think it's reasonable to assume that most people just want to get to where they're going in the least amount of time safely. Not necessarily in a mad rush. Not race day at the Daytona 500, but relatively quickly. And yet, if you try to observe the simple 3 second rule and leave a nice gap between you and the car in front of you, you get stepped on. Not cut off, but you'll get bumped back, again and again and again.

Traffic out here is like a line at the bank. Would you, in person, stand in line at the bank and let anyone cut in front of you simply because you didn't take a step or two forward when the person in front

Apparently, a lot of people think that their vehicles only have "stop" and "go" modes, without multiple different levels of coasting and slowing down without the brakes for those minor adjustments. I'll take my car out of overdrive quite often to adjust just a bit, rather than hitting the brakes needlessly. That way when I hit my brakes, I actually mean it. Brake lights = slowing down non-trivially. I wish more people did that.

Preach it brother! I had a friend visiting from Europe a few years back and she'd go for the "oh shit!" bar whenever I would pass on the right, a maneuver that is more or less required in Florida because there's inevitably some 80 year old doing 5 miles an hour under the speed limit sitting in the left lane. Every few months down there you get some story in the newspaper either about some elderly drivers who got confused and got on the Interstate going the wrong way or about some legally blind driver who took the driving test (!) 25 times and then finally getting a license.

I'm not just pointing a finger at the elderly either. Stupidity abounds across the age spectrum. A lot of people seem to feel that operating a 2 ton vehicle is "free time" and not time they need to spend actually paying attention to the task at hand, which is OPERATING A 2 TON VEHICLE! Anything that distracts you from doing that and doing it well should be grounds for a ticket. And the training to drill that point home should be required as part of the license process.

I tend to do this during heavy traffic. I drive a manual, and shifting in and out of gear over and over gets old real fast. The trick is just to not care. It's hard, I know. Give yourself more than 2 seconds. I tend to go by car lengths, just because traffic conditions vary. In slow traffic (5-25 mph) I'll try to maintain a 10-car-length space or so. This doesn't mean I'm constantly braking and accelerating; if I wanted to do that, I'd just tailgate the guy in front of me like everyone else. If someone cuts in in front of you, let them. They're probably just going to speed up and slam on their brakes when they get to the car in front of you. You can slightly adjust your speed by releasing the gas pedal a little to maintain your space. There's no real hurry -- you have 9 car lengths worth of time to do this. If traffic is heavy and slow enough, you end up with people to the right and left (and behind) you who aren't "cheaters," and people tend to stop cutting in front of you so much. But even if they do, the magic phrase is: "who cares?"

Yeah, but when you start giving enough room between you and the car ahead, an idiot besides you speeds up and steals your place:-/

True. That rudeness is endemic and remains as painful today as it did in days gone by is also true. The alternative to not maintaining a safe distance to prevent others from merging into their own unsafe distance is... wait for it... driving too close. Sounds more like a comedy of errors than a strategy, doesn't it?

What should offer you satisfaction is that you're adopting the only sane approach. This benefits not only you, but also everyone else on the road. So each time it happens, slow down a bit, and don't give it any more thought than it deserves. Your actions will have a negligible effect on your own travel time, and that of everyone behind you. The added bonus for losing giving up some of your ego is that you'll get to where you're going safely, and in a better mood.

is not self-driving cars, but public transportation. And if you need something heavy moved, have it delivered or rent a pickup for a day. Owning and operating cars make sense where the population density falls below a certain threshhold, say in the country, but in sub/urban spaces, which is what we're talking about here, there's no good reason to use the car as the solution to personal transportation.

And that's just for logistical reasons. When you consider the cost to the environment, the justification weakens more. When you consider the cost to our foreign policy and national security in being dependent upon other countries for oil, the justification weakens still more. When you consider the sheer hassle and productivity lost to accidents, finding parking, breakdowns, time lost sitting in traffic, and aggravation of driving (people cutting you off, getting stuck behind a slow poke, etc), the justification almost evaporates. And when you think about what the $15K you drop on a car and the $5K/yr. worth of insurance, gas, parking, and repairs you have to put into it to keep it running, and the reality that the value of the thing itself loses half its value every year, versus what that money could do for you if you even put it into an index fund, then financially it's the last nail in the coffin for the justification of owning a car.

Is it idiotic to do what is best of your own self interest? The answer is more complicated than you might think, it depends upon what your adversaries do. This situation sounds very similar to the prisoners dilemma [wikipedia.org] or indeed any other situation where individuals acting in their own self interest collectively reach a worse result (for each of them individually) than would otherwise have occurred if they cooperated instead of taking the greedy approach (i.e the Tragedy of the Commons [wikipedia.org]).

Because people aren't smart enough to do that. You see the same people who will do 40mph up to a red light just to sit there. I on the other hand try to guess when the light is going to change and just brake a little, so that I hit the intersection just when it turns green and still have some momentum left saving myself some fuel and time. People just don't think about what they're doing, and if you explain it to them they don't listen.

The real problem seems to be that it's in everyone's best interest to not be jerks, but if everyone's not a jerk, any selfish individual can be a jerk and do (marginally) better than everyone else. The situation with traffic we have now is the worst scenario of all... Everyone's being a jerk, causing the worst outcomes for all.

If we were to automate cars the left two lanes of most freeways would be vacant. I commute on a motorcycle and have observed the ins and outs of traffic (what with my life depending on it and all) and those people are seriously F-ed in the head, I propose a formula for the mathematicians to try that will give the real reason for traffic: Norman Person + oversized vehicle + minor competition = Total Fuckwad. Basically these people should be killed for our national security: 3,620 people die monthly due to automobile accidents, while 2,974 died from 9/11.

So? Let them have it, and then give them enough space so that you won't have to brake. You'll *STILL* get through there faster than if you rode the ass of the guy in front of you and caused a traffic jam.

I have to admit, vision is probably my biggest reason for SUV bashing. I don't care about that "saving the planet" bullshit. I don't care about somebody else's penis-size compensation, or their deluded feelings about safety. I don't care how much someone else spends on gasoline. Those aren't my problems.

I do care about being able to see. I can't see through large vehicles. It makes me nervous as hell, whether following them, or trying to make a left turn when one of those fuckers is in the opposing left-turn lane.

Either get your huge piece of shit off my street, or build it out of untinted glass, goddammit.

Courtesy of a course I audited at MIT on discrete mathematics, about 20 years ago, which included a fascinating section on the mathematics of traffic...

(1) They're not really standing waves, which are composed of traveling waves going both forward and backward (and waves can't propagate forward in traffic). They're ordinary traveling waves. The best analogy is to the flow of a compressible gas in a pipe. You can easily get strong shock waves at various densities and flow rates when you introduce obstructions or change the flow rate at various branch points.

Part of the problem in our expectations is that we (unreasonably) expect traffic flow to be more like the flow of an incompressible fluid like water, where, generally speaking, more pressure simply equals faster flow. It's the presence of compressibility that makes gas flow in certain critical regions much more complicated than water flow, so that, for example, an increase in pressure (e.g. an increase in cars entering at a given on-ramp, or a constriction due to an accident) can result in drastic decreases in flow. The compressibility comes about in traffic because the density of cars is quite variable.

(2) Along those lines, the density per se -- the space between the cars -- really has very little to do with the peculiarities of traffic. It's the fact that the density can change locally which makes the "car gas" compressible, and allows for density waves (traffic jams, stop-n-go traffic, etc.).

But the reason the density changes locally is not because people don't leave enough space between their car and the car ahead, but because of human reaction time. If the car spacing (i.e. density) changes here at time t, human reaction time means it cannot propagate very fast -- it will change there at some time t' significantly later than t. That is, a density wave must propagate. Under the right conditions, it's quite easy for such a density wave to grow in amplitude as it goes. Hence, a very small initial perturbation in the density -- one driver slamming on the brakes -- can grow much larger as it propagates, so that at some distance away large numbers of cars must come to a halt.

The only real solution is to make the car "gas" much less compressible, and that requires greatly raising the speed at which density fluctuations can propagate, in other words, tremendously shortening the time it takes for cars to respond to slight changes in spacing. Presumably, that suggests computer control of cars.

Yes, I thought for any engineer, student of fluid dynamics, this was a known. Isn't this part of bernoulli's equations?

I usually try to buffer the stoppage effect for people behind me by slowing sufficiently ahead so as to not stop or remain slightly faster than the person in front of me, so that when they speed up, I don't use as much force and energy getting back up to speed and that section of the wave will move slightly faster than it currently was.

I'll take my car out of overdrive quite often to adjust just a bit, rather than hitting the brakes needlessly.

That's one of the main reasons I drive a manual transmission: I can adjust my speed without having to use my brakes. Makes the brake pads last longer too.

I swear, we could reduce the amount of idiots on the road by making two things mandatory for new drivers: (1) learn on a manual transmission first, and (2) make parallel parking part of the driver's test (again).

As a recent arrival in America from Australia, I find myself amazed by how seldom people use their turn signals. Almost never. Annoys the hell out of me, and my other pet peeve: when I signal that I'm going to overtake a slow vehicle. Turn signal on, wait for lane beside me to clear, only to have some fucking clown decide he wants to get past both of us and shoot right out from behind me, no signal, nearly rear ending me in the process, and then has the audacity to blow his horn at me. Gah.

People tailgate and then apply their brakes. This is the proximate cause of the whole thing.

Think of traffic like a rubber band. It has nothing to do with that at all. It's a problem, but not that bad.

The real issue is that the slower you drive, the less distance required to leave between you and the next vehicle. At a stop, the distance is 0, and at 100 km/h it's probably three car lengths - maybe 60ft or so.

Graph it out and it's probably quite a linear relationship - at 50 km/h 30ft is probably acceptable.

So you have a bunch of people driving 60ft apart on the highway, and a bunch of cars merge in. The cars at 60ft now need to slow down to increase their distance. Once said car slows down, the car behind has to slow down too, except someone is merging in front of him, and so on.

That goes for stop lights too. You have a bunch of people driving 30ft apart come to a stop. They all don't start instantly, they have to wait for bonus ft. When you're the 10th car, you need an extra 300ft to get going.

It's not about the roads. It's about the people.The dickheads that has to press into an intersection even though he sees that it's blocked. It's that dickhead who won't let people change lanes. The dickhead who is blocked from changing lanes and has to stop in the middle of the road to change lanes *now*. That dickhead who drives under the speed limit in the left lane.

When we get cars that drive themselves, we can fit five times more cars on our current roads, with no traffic jams.

And public transport = fail. I do not know where you are from, but I do know the public transport in Amsterdam technically works. It's the people fucking it all up again.The subway next to me should go about every 5 or 10 minutes in the mornings. However at 8-8:30 there goes no subways past. At 8:30 precisely there comes one to the station. The door open and people explode out due that the car is so overfilled. The subway passes, and in the next two minutes 3 or 4 almost empty ones pass.

Very bright accompanied by a rapidly flashing strobe: indicates the car is braking maximally; antilock braking system is fully engaged. (At times like this, the car should do everything possible to get the attention of other drivers.)

And indicates that you will have some serious traffic accidents and deaths when some epileptics (including people who don't know they have it) go into seizures from those rapidly flashing strobes.

It seems to me that the major deficiency is in your public transport system. Fixing / building well-interconnected rail, bus and bicycle routes would mitigate your traffic problems in a much more sensible way than somehow modifying your highway system, car technology or road laws.

Something to keep in mind is that not everyone is going to the stop light like you are. They may be making a turn beforehand. Or they could be turning left, and where I live the left turn arrow turns green before the light turns green, so if the light is red and someone wants to turn left, they do want to get there before it changes.

I still do as you do, but I do try to gauge the intentions of cars behind me. If they have their signal on I won't block their way, but if they are one of those that don't signal, well, screw them.